A064827 -id:A064827 - cp's OEIS frontend

A046827 Numbers k such that k^2 contains all the digits of k with the same or higher multiplicity.

0, 1, 5, 6, 10, 11, 25, 27, 50, 60, 63, 64, 74, 76, 95, 96, 100, 101, 105, 110, 125, 139, 142, 205, 250, 255, 261, 270, 275, 277, 278, 285, 305, 364, 371, 376, 405, 421, 441, 463, 472, 493, 497, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 523, 524, 525

Offset: 1

David W. Wilson

10^n is a term for all n. - Amarnath Murthy, Aug 03 2005

			27 is a term as 27^2 = 729 contains 2 and 7.
255 is a term as 255^2 = 65025 which contains the digits 2,5,5. 502 is a term as 502^2 = 252004 which contains 5, 0, 2.

Cf. A046829, A029772.

Cf. A064827 (essentially the same).

isA046827 := proc(n) local dgsn,dgsnsq,multsn,multsn2,o,i ;
dgsn := sort(convert(n,base,10)) ;
dgsnsq := sort(convert(n^2,base,10)) ;
multsn := [seq(0,i=0..9) ] ;
multsn2 := [seq(0,i=0..9) ] ; for i from 1 to nops(dgsn) do o := op(1+op(i,dgsn),multsn) ; multsn := subsop( 1+op(i,dgsn)=o+1,multsn ) ; od: for i from 1 to nops(dgsnsq) do o := op(1+op(i,dgsnsq),multsn2) ; multsn2 := subsop( 1+op(i,dgsnsq)=o+1,multsn2 ) ; od: for i from 1 to 10 do if op(i,multsn2) < op(i,multsn) then RETURN(false) ; fi ; od: RETURN(true) ; end: for n from 1 to 700 do if isA046827(n) then printf("%d,",n) ; fi ; od; # R. J. Mathar, Feb 11 2008

Join[{0}, Select[Range[525], Count[Table[DigitCount[#^2, 10, k] - DigitCount[#, 10, k], {k, Union[IntegerDigits[#]]}], ?Negative] == 0 &]] (* _Jayanta Basu, Jun 29 2013 *)

from itertools import count, islice
from collections import Counter
def A046827_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda k:Counter(str(k))<=Counter(str(k**2)),count(max(startvalue,0)))
A046827_list = list(islice(A046827_gen(),20)) # Chai Wah Wu, Apr 03 2023

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

A326418 Nonnegative numbers k such that, in decimal representation, the subsequence of digits of k^2 occupying an odd position is equal to the digits of k.

Original entry on oeis.org

0, 1, 5, 6, 10, 11, 50, 60, 76, 100, 105, 110, 500, 501, 505, 506, 600, 605, 756, 760, 826, 1000, 1001, 1050, 1100, 5000, 5010, 5050, 5060, 5941, 6000, 6050, 7560, 7600, 8260, 10000, 10005, 10010, 10500, 10505, 11000, 12731

Offset: 1

Riccardo Pietro Giovambattista Mazzei and Matteo Albanese, Sep 28 2019

If k is in the sequence then so is 10*k. - David A. Corneth, Sep 29 2019

No term starts with the digit 2. - Chai Wah Wu, Apr 04 2023

			5^2 = 25, whose first digit is 5, hence 5 is a term of the sequence.
11^2 = 121, whose first and third digit are (1, 1), hence 11 is a term of the sequence.
756^2 = 571536, whose digits in odd positions - starting from the least significant one - are (7, 5, 6), hence 756 is a term of the sequence.

David A. Corneth, Table of n, a(n) for n = 1..2361 (terms < 10^15; first 485 terms from Charles R Greathouse IV)

Subsequence of A008851, A029772, A046829, A064827, A052212, A189056.

Select[Range[0, 13000], Reverse@ #[[-Range[1, Length@ #, 2]]] &@ IntegerDigits[#^2] === IntegerDigits[#] &] (* Michael De Vlieger, Oct 06 2019 *)

isok(n) = my(d=Vecrev(digits(n^2))); fromdigits(Vecrev(vector((#d+1)\2, k, d[2*k-1]))) == n; \\ Michel Marcus, Oct 01 2019

def ok(n): s = str(n*n); return n == int("".join(s[1-len(s)%2::2]))
print(list(filter(ok, range(13000)))) # Michael S. Branicky, Sep 10 2021

A178246 Numbers m such that all digits of m, including repetitions, occur among the digits of 2^m.

Original entry on oeis.org

6, 10, 14, 17, 21, 25, 27, 28, 29, 30, 31, 35, 36, 37, 39, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117

Offset: 1

Michel Lagneau, Dec 20 2010

The sequence shows subsets of consecutive numbers.

153 is assumed to be the largest integer missing in this sequence. - Alois P. Heinz, Jan 28 2022

			17 is a term because the digits 1 and 7 are included in the number 2^17 = 131072;
3 is not a term because the digit 3 is not in the number 2^3 = 8.
33 is not a term because 2^33 = 8589934592 does not have 2 digits 3.
153 is not in the sequence because the digit 3 is not in the number 2^153 = 11417981541647679048466287755595961091061972992.

Michel Marcus, Table of n, a(n) for n = 1..962

Cf. A000079, A064827, A032740.

Reap[Do[a = DigitCount[2^n]; b = DigitCount[n]; If[Min[a-b] >= 0, Sow[n]], {n, 10^3}]][[2, 1]]

isok(m) = {my(d=digits(m), dd=digits(2^m)); for (i=0, 9, if (#select(x->(x==i), d) > #select(x->(x==i), dd), return (0));); return(1);} \\ Michel Marcus, Jan 28 2022

Name clarified by Michel Marcus, Jan 30 2022

A178223 Numbers n such that the digits of n are also digits of n! (counting multiplicity).

Original entry on oeis.org

1, 2, 4, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112

Offset: 1

Michel Lagneau, Dec 20 2010

			23 is in the sequence because 2 and 3 are digits of 23! = 25852016738884976640000 ;
28 is not in the sequence because 2 is not a digit of 28! = 304888344611713860501504000000.
11 is not in the sequence because 11! = 39916800 contains only a single "1".

Cf. A064827

Reap[Do[a = DigitCount[n!]; b = DigitCount[n]; If[Min[a-b] >= 0, Sow[n]], {n, 1, 10^3}]][[2, 1]]

cp's OEIS Frontend

A046827 Numbers k such that k^2 contains all the digits of k with the same or higher multiplicity.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A326418 Nonnegative numbers k such that, in decimal representation, the subsequence of digits of k^2 occupying an odd position is equal to the digits of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A178246 Numbers m such that all digits of m, including repetitions, occur among the digits of 2^m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A178223 Numbers n such that the digits of n are also digits of n! (counting multiplicity).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica